// The following is adapted from fdlibm (http://www.netlib.org/fdlibm),
//
// ====================================================
-// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+// Copyright (C) 1993-2004 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// modified significantly by Google Inc.
// Copyright 2014 the V8 project authors. All rights reserved.
//
-// The following is a straightforward translation of fdlibm routines for
-// sin, cos, and tan, by Raymond Toy (rtoy@google.com).
+// The following is a straightforward translation of fdlibm routines
+// by Raymond Toy (rtoy@google.com).
+// Double constants that do not have empty lower 32 bits are found in fdlibm.cc
+// and exposed through kMath as typed array. We assume the compiler to convert
+// from decimal to binary accurately enough to produce the intended values.
+// kMath is initialized to a Float64Array during genesis and not writable.
+var kMath;
-var kTrig; // Initialized to a Float64Array during genesis and is not writable.
-
-const INVPIO2 = kTrig[0];
-const PIO2_1 = kTrig[1];
-const PIO2_1T = kTrig[2];
-const PIO2_2 = kTrig[3];
-const PIO2_2T = kTrig[4];
-const PIO2_3 = kTrig[5];
-const PIO2_3T = kTrig[6];
-const PIO4 = kTrig[32];
-const PIO4LO = kTrig[33];
+const INVPIO2 = kMath[0];
+const PIO2_1 = kMath[1];
+const PIO2_1T = kMath[2];
+const PIO2_2 = kMath[3];
+const PIO2_2T = kMath[4];
+const PIO2_3 = kMath[5];
+const PIO2_3T = kMath[6];
+const PIO4 = kMath[32];
+const PIO4LO = kMath[33];
// Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For
// precision, r is returned as two values y0 and y1 such that r = y0 + y1
// sin(x) = X + (S1*X + (X *(r-Y/2)+Y))
//
macro KSIN(x)
-kTrig[7+x]
+kMath[7+x]
endmacro
macro RETURN_KERNELSIN(X, Y, SIGN)
// thus, reducing the rounding error in the subtraction.
//
macro KCOS(x)
-kTrig[13+x]
+kMath[13+x]
endmacro
macro RETURN_KERNELCOS(X, Y, SIGN)
}
endmacro
+
// kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
// Input x is assumed to be bounded by ~pi/4 in magnitude.
// Input y is the tail of x.
// and will cause incorrect results.
//
macro KTAN(x)
-kTrig[19+x]
+kMath[19+x]
endmacro
function KernelTan(x, y, returnTan) {
REMPIO2(x);
return KernelTan(y0, y1, (n & 1) ? -1 : 1);
}
+
+// ES6 draft 09-27-13, section 20.2.2.20.
+// Math.log1p
+//
+// Method :
+// 1. Argument Reduction: find k and f such that
+// 1+x = 2^k * (1+f),
+// where sqrt(2)/2 < 1+f < sqrt(2) .
+//
+// Note. If k=0, then f=x is exact. However, if k!=0, then f
+// may not be representable exactly. In that case, a correction
+// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
+// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
+// and add back the correction term c/u.
+// (Note: when x > 2**53, one can simply return log(x))
+//
+// 2. Approximation of log1p(f).
+// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+// = 2s + s*R
+// We use a special Reme algorithm on [0,0.1716] to generate
+// a polynomial of degree 14 to approximate R The maximum error
+// of this polynomial approximation is bounded by 2**-58.45. In
+// other words,
+// 2 4 6 8 10 12 14
+// R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
+// (the values of Lp1 to Lp7 are listed in the program)
+// and
+// | 2 14 | -58.45
+// | Lp1*s +...+Lp7*s - R(z) | <= 2
+// | |
+// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+// In order to guarantee error in log below 1ulp, we compute log
+// by
+// log1p(f) = f - (hfsq - s*(hfsq+R)).
+//
+// 3. Finally, log1p(x) = k*ln2 + log1p(f).
+// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+// Here ln2 is split into two floating point number:
+// ln2_hi + ln2_lo,
+// where n*ln2_hi is always exact for |n| < 2000.
+//
+// Special cases:
+// log1p(x) is NaN with signal if x < -1 (including -INF) ;
+// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
+// log1p(NaN) is that NaN with no signal.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Constants:
+// Constants are found in fdlibm.cc. We assume the C++ compiler to convert
+// from decimal to binary accurately enough to produce the intended values.
+//
+// Note: Assuming log() return accurate answer, the following
+// algorithm can be used to compute log1p(x) to within a few ULP:
+//
+// u = 1+x;
+// if (u==1.0) return x ; else
+// return log(u)*(x/(u-1.0));
+//
+// See HP-15C Advanced Functions Handbook, p.193.
+//
+const LN2_HI = kMath[34];
+const LN2_LO = kMath[35];
+const TWO54 = kMath[36];
+const TWO_THIRD = kMath[37];
+macro KLOG1P(x)
+(kMath[38+x])
+endmacro
+
+function MathLog1p(x) {
+ x = x * 1; // Convert to number.
+ var hx = %_DoubleHi(x);
+ var ax = hx & 0x7fffffff;
+ var k = 1;
+ var f = x;
+ var hu = 1;
+ var c = 0;
+ var u = x;
+
+ if (hx < 0x3fda827a) {
+ // x < 0.41422
+ if (ax >= 0x3ff00000) { // |x| >= 1
+ if (x === -1) {
+ return -INFINITY; // log1p(-1) = -inf
+ } else {
+ return NAN; // log1p(x<-1) = NaN
+ }
+ } else if (ax < 0x3c900000) {
+ // For |x| < 2^-54 we can return x.
+ return x;
+ } else if (ax < 0x3e200000) {
+ // For |x| < 2^-29 we can use a simple two-term Taylor series.
+ return x - x * x * 0.5;
+ }
+
+ if ((hx > 0) || (hx <= -0x402D413D)) { // (int) 0xbfd2bec3 = -0x402d413d
+ // -.2929 < x < 0.41422
+ k = 0;
+ }
+ }
+
+ // Handle Infinity and NAN
+ if (hx >= 0x7ff00000) return x;
+
+ if (k !== 0) {
+ if (hx < 0x43400000) {
+ // x < 2^53
+ u = 1 + x;
+ hu = %_DoubleHi(u);
+ k = (hu >> 20) - 1023;
+ c = (k > 0) ? 1 - (u - x) : x - (u - 1);
+ c = c / u;
+ } else {
+ hu = %_DoubleHi(u);
+ k = (hu >> 20) - 1023;
+ }
+ hu = hu & 0xfffff;
+ if (hu < 0x6a09e) {
+ u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u)); // Normalize u.
+ } else {
+ ++k;
+ u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u)); // Normalize u/2.
+ hu = (0x00100000 - hu) >> 2;
+ }
+ f = u - 1;
+ }
+
+ var hfsq = 0.5 * f * f;
+ if (hu === 0) {
+ // |f| < 2^-20;
+ if (f === 0) {
+ if (k === 0) {
+ return 0.0;
+ } else {
+ return k * LN2_HI + (c + k * LN2_LO);
+ }
+ }
+ var R = hfsq * (1 - TWO_THIRD * f);
+ if (k === 0) {
+ return f - R;
+ } else {
+ return k * LN2_HI - ((R - (k * LN2_LO + c)) - f);
+ }
+ }
+
+ var s = f / (2 + f);
+ var z = s * s;
+ var R = z * (KLOG1P(0) + z * (KLOG1P(1) + z *
+ (KLOG1P(2) + z * (KLOG1P(3) + z *
+ (KLOG1P(4) + z * (KLOG1P(5) + z * KLOG1P(6)))))));
+ if (k === 0) {
+ return f - (hfsq - s * (hfsq + R));
+ } else {
+ return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f);
+ }
+}
+
+// ES6 draft 09-27-13, section 20.2.2.14.
+// Math.expm1
+// Returns exp(x)-1, the exponential of x minus 1.
+//
+// Method
+// 1. Argument reduction:
+// Given x, find r and integer k such that
+//
+// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
+//
+// Here a correction term c will be computed to compensate
+// the error in r when rounded to a floating-point number.
+//
+// 2. Approximating expm1(r) by a special rational function on
+// the interval [0,0.34658]:
+// Since
+// r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
+// we define R1(r*r) by
+// r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
+// That is,
+// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+// = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
+// We use a special Remes algorithm on [0,0.347] to generate
+// a polynomial of degree 5 in r*r to approximate R1. The
+// maximum error of this polynomial approximation is bounded
+// by 2**-61. In other words,
+// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+// where Q1 = -1.6666666666666567384E-2,
+// Q2 = 3.9682539681370365873E-4,
+// Q3 = -9.9206344733435987357E-6,
+// Q4 = 2.5051361420808517002E-7,
+// Q5 = -6.2843505682382617102E-9;
+// (where z=r*r, and the values of Q1 to Q5 are listed below)
+// with error bounded by
+// | 5 | -61
+// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
+// | |
+//
+// expm1(r) = exp(r)-1 is then computed by the following
+// specific way which minimize the accumulation rounding error:
+// 2 3
+// r r [ 3 - (R1 + R1*r/2) ]
+// expm1(r) = r + --- + --- * [--------------------]
+// 2 2 [ 6 - r*(3 - R1*r/2) ]
+//
+// To compensate the error in the argument reduction, we use
+// expm1(r+c) = expm1(r) + c + expm1(r)*c
+// ~ expm1(r) + c + r*c
+// Thus c+r*c will be added in as the correction terms for
+// expm1(r+c). Now rearrange the term to avoid optimization
+// screw up:
+// ( 2 2 )
+// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
+// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
+// ( )
+//
+// = r - E
+// 3. Scale back to obtain expm1(x):
+// From step 1, we have
+// expm1(x) = either 2^k*[expm1(r)+1] - 1
+// = or 2^k*[expm1(r) + (1-2^-k)]
+// 4. Implementation notes:
+// (A). To save one multiplication, we scale the coefficient Qi
+// to Qi*2^i, and replace z by (x^2)/2.
+// (B). To achieve maximum accuracy, we compute expm1(x) by
+// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
+// (ii) if k=0, return r-E
+// (iii) if k=-1, return 0.5*(r-E)-0.5
+// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
+// else return 1.0+2.0*(r-E);
+// (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
+// (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
+// (vii) return 2^k(1-((E+2^-k)-r))
+//
+// Special cases:
+// expm1(INF) is INF, expm1(NaN) is NaN;
+// expm1(-INF) is -1, and
+// for finite argument, only expm1(0)=0 is exact.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Misc. info.
+// For IEEE double
+// if x > 7.09782712893383973096e+02 then expm1(x) overflow
+//
+const KEXPM1_OVERFLOW = kMath[45];
+const INVLN2 = kMath[46];
+macro KEXPM1(x)
+(kMath[47+x])
+endmacro
+
+function MathExpm1(x) {
+ x = x * 1; // Convert to number.
+ var y;
+ var hi;
+ var lo;
+ var k;
+ var t;
+ var c;
+
+ var hx = %_DoubleHi(x);
+ var xsb = hx & 0x80000000; // Sign bit of x
+ var y = (xsb === 0) ? x : -x; // y = |x|
+ hx &= 0x7fffffff; // High word of |x|
+
+ // Filter out huge and non-finite argument
+ if (hx >= 0x4043687a) { // if |x| ~=> 56 * ln2
+ if (hx >= 0x40862e42) { // if |x| >= 709.78
+ if (hx >= 0x7ff00000) {
+ // expm1(inf) = inf; expm1(-inf) = -1; expm1(nan) = nan;
+ return (x === -INFINITY) ? -1 : x;
+ }
+ if (x > KEXPM1_OVERFLOW) return INFINITY; // Overflow
+ }
+ if (xsb != 0) return -1; // x < -56 * ln2, return -1.
+ }
+
+ // Argument reduction
+ if (hx > 0x3fd62e42) { // if |x| > 0.5 * ln2
+ if (hx < 0x3ff0a2b2) { // and |x| < 1.5 * ln2
+ if (xsb === 0) {
+ hi = x - LN2_HI;
+ lo = LN2_LO;
+ k = 1;
+ } else {
+ hi = x + LN2_HI;
+ lo = -LN2_LO;
+ k = -1;
+ }
+ } else {
+ k = (INVLN2 * x + ((xsb === 0) ? 0.5 : -0.5)) | 0;
+ t = k;
+ // t * ln2_hi is exact here.
+ hi = x - t * LN2_HI;
+ lo = t * LN2_LO;
+ }
+ x = hi - lo;
+ c = (hi - x) - lo;
+ } else if (hx < 0x3c900000) {
+ // When |x| < 2^-54, we can return x.
+ return x;
+ } else {
+ // Fall through.
+ k = 0;
+ }
+
+ // x is now in primary range
+ var hfx = 0.5 * x;
+ var hxs = x * hfx;
+ var r1 = 1 + hxs * (KEXPM1(0) + hxs * (KEXPM1(1) + hxs *
+ (KEXPM1(2) + hxs * (KEXPM1(3) + hxs * KEXPM1(4)))));
+ t = 3 - r1 * hfx;
+ var e = hxs * ((r1 - t) / (6 - x * t));
+ if (k === 0) { // c is 0
+ return x - (x*e - hxs);
+ } else {
+ e = (x * (e - c) - c);
+ e -= hxs;
+ if (k === -1) return 0.5 * (x - e) - 0.5;
+ if (k === 1) {
+ if (x < -0.25) return -2 * (e - (x + 0.5));
+ return 1 + 2 * (x - e);
+ }
+
+ if (k <= -2 || k > 56) {
+ // suffice to return exp(x) + 1
+ y = 1 - (e - x);
+ // Add k to y's exponent
+ y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
+ return y - 1;
+ }
+ if (k < 20) {
+ // t = 1 - 2^k
+ t = %_ConstructDouble(0x3ff00000 - (0x200000 >> k), 0);
+ y = t - (e - x);
+ // Add k to y's exponent
+ y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
+ } else {
+ // t = 2^-k
+ t = %_ConstructDouble((0x3ff - k) << 20, 0);
+ y = x - (e + t);
+ y += 1;
+ // Add k to y's exponent
+ y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
+ }
+ }
+ return y;
+}
+
+
+// ES6 draft 09-27-13, section 20.2.2.30.
+// Math.sinh
+// Method :
+// mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
+// 1. Replace x by |x| (sinh(-x) = -sinh(x)).
+// 2.
+// E + E/(E+1)
+// 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
+// 2
+//
+// 22 <= x <= lnovft : sinh(x) := exp(x)/2
+// lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
+// ln2ovft < x : sinh(x) := x*shuge (overflow)
+//
+// Special cases:
+// sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
+// only sinh(0)=0 is exact for finite x.
+//
+const KSINH_OVERFLOW = kMath[52];
+const TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half
+const LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half
+
+function MathSinh(x) {
+ x = x * 1; // Convert to number.
+ var h = (x < 0) ? -0.5 : 0.5;
+ // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
+ var ax = MathAbs(x);
+ if (ax < 22) {
+ // For |x| < 2^-28, sinh(x) = x
+ if (ax < TWO_M28) return x;
+ var t = MathExpm1(ax);
+ if (ax < 1) return h * (2 * t - t * t / (t + 1));
+ return h * (t + t / (t + 1));
+ }
+ // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
+ if (ax < LOG_MAXD) return h * MathExp(ax);
+ // |x| in [log(maxdouble), overflowthreshold]
+ // overflowthreshold = 710.4758600739426
+ if (ax <= KSINH_OVERFLOW) {
+ var w = MathExp(0.5 * ax);
+ var t = h * w;
+ return t * w;
+ }
+ // |x| > overflowthreshold or is NaN.
+ // Return Infinity of the appropriate sign or NaN.
+ return x * INFINITY;
+}
+
+
+// ES6 draft 09-27-13, section 20.2.2.12.
+// Math.cosh
+// Method :
+// mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
+// 1. Replace x by |x| (cosh(x) = cosh(-x)).
+// 2.
+// [ exp(x) - 1 ]^2
+// 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
+// 2*exp(x)
+//
+// exp(x) + 1/exp(x)
+// ln2/2 <= x <= 22 : cosh(x) := -------------------
+// 2
+// 22 <= x <= lnovft : cosh(x) := exp(x)/2
+// lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
+// ln2ovft < x : cosh(x) := huge*huge (overflow)
+//
+// Special cases:
+// cosh(x) is |x| if x is +INF, -INF, or NaN.
+// only cosh(0)=1 is exact for finite x.
+//
+const KCOSH_OVERFLOW = kMath[52];
+
+function MathCosh(x) {
+ x = x * 1; // Convert to number.
+ var ix = %_DoubleHi(x) & 0x7fffffff;
+ // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
+ if (ix < 0x3fd62e43) {
+ var t = MathExpm1(MathAbs(x));
+ var w = 1 + t;
+ // For |x| < 2^-55, cosh(x) = 1
+ if (ix < 0x3c800000) return w;
+ return 1 + (t * t) / (w + w);
+ }
+ // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
+ if (ix < 0x40360000) {
+ var t = MathExp(MathAbs(x));
+ return 0.5 * t + 0.5 / t;
+ }
+ // |x| in [22, log(maxdouble)], return half*exp(|x|)
+ if (ix < 0x40862e42) return 0.5 * MathExp(MathAbs(x));
+ // |x| in [log(maxdouble), overflowthreshold]
+ if (MathAbs(x) <= KCOSH_OVERFLOW) {
+ var w = MathExp(0.5 * MathAbs(x));
+ var t = 0.5 * w;
+ return t * w;
+ }
+ if (NUMBER_IS_NAN(x)) return x;
+ // |x| > overflowthreshold.
+ return INFINITY;
+}
projects / platform / framework / web / crosswalk.git / blobdiff